Model Deforms When Trying to Animate
Tweaked three things and got it to work.



Tris to Quads

The first thing I did was Tab into Edit Mode and key Alt + J to convert Tris to Quads. This is a form of retopologizing that is highly recommended. Triangles are not recommended when modeling.

Remove Doubles

I keyed W to open the Specials menu and selected Remove Doubles. This got rid of excess duplicate vertices that were messing with your Weight Painting. Of course, this destroyed the shading, so I added an Edge Split modifier in the Properties panel > Object modifiers tab.



Parenting

After all the changes to the mesh, I redid the parenting. I selected the mesh, then selected the armature, keyed Ctrl + P, and chose Armature Deform > With Automatic Weights.



Here is the fixed .blend file: